Embedded newform 287.3.p.a.146.6

• Label: 287.3.p.a.146.6
• Level: $287$
• Weight: $3$
• Character: 287.146
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.p (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			146.6
Character	\(\chi\)	\(=\)	287.146
Dual form			287.3.p.a.230.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07676 - 3.31392i) q^{2} +5.20007 q^{3} +(-6.58656 + 4.78542i) q^{4} +(3.83595 + 5.27973i) q^{5} +(-5.59921 - 17.2326i) q^{6} +(2.29624 - 6.61266i) q^{7} +(11.6747 + 8.48214i) q^{8} +18.0408 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{2} - 110 q^{4} - 5 q^{7} + 22 q^{8} + 552 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07676	−	3.31392i	−0.538378	−	1.65696i	−0.736234	−	0.676727i	\(-0.763398\pi\)
							0.197856	−	0.980231i	\(-0.436602\pi\)
\(3\)	5.20007			1.73336			0.866679	−	0.498867i	\(-0.166250\pi\)
							0.866679	+	0.498867i	\(0.166250\pi\)
\(5\)	3.83595	+	5.27973i	0.767189	+	1.05595i	0.996582	+	0.0826117i	\(0.0263261\pi\)
							−0.229392	+	0.973334i	\(0.573674\pi\)
\(7\)	2.29624	−	6.61266i	0.328034	−	0.944666i
							\(11\)	−5.80125	+	7.98474i	−0.527387	+	0.725885i	−0.986729	−	0.162374i	\(-0.948085\pi\)
0.459343	+	0.888259i	\(0.348085\pi\)
\(13\)	2.67817	+	8.24257i	0.206013	+	0.634044i	0.999670	+	0.0256783i	\(0.00817457\pi\)
							−0.793657	+	0.608366i	\(0.791825\pi\)
\(17\)	19.9239	+	14.4756i	1.17199	+	0.851503i	0.991246	−	0.132026i	\(-0.0421483\pi\)
							0.180748	+	0.983530i	\(0.442148\pi\)
\(19\)	10.7325	−	33.0311i	0.564866	−	1.73848i	−0.103482	−	0.994631i	\(-0.532999\pi\)
							0.668349	−	0.743848i	\(-0.267001\pi\)
\(23\)	−7.77370	−	23.9250i	−0.337987	−	1.04022i	−0.965232	−	0.261395i	\(-0.915817\pi\)
							0.627245	−	0.778822i	\(-0.284183\pi\)
\(29\)	−20.1821	−	27.7783i	−0.695934	−	0.957871i	−0.999986	−	0.00521568i	\(-0.998340\pi\)
							0.304052	−	0.952655i	\(-0.401660\pi\)
\(31\)	−12.9292	+	17.7955i	−0.417071	+	0.574049i	−0.964925	−	0.262525i	\(-0.915445\pi\)
							0.547854	+	0.836574i	\(0.315445\pi\)
\(37\)	−29.4738	+	21.4140i	−0.796589	+	0.578756i	−0.909912	−	0.414802i	\(-0.863851\pi\)
							0.113322	+	0.993558i	\(0.463851\pi\)
\(41\)	−40.6070	−	5.66318i	−0.990415	−	0.138126i
							\(43\)	2.00850	+	6.18152i	0.0467092	+	0.143756i	0.971691	−	0.236255i	\(-0.0759201\pi\)
−0.924982	+	0.380011i	\(0.875920\pi\)
\(47\)	4.62652	+	14.2390i	0.0984366	+	0.302957i	0.988134	−	0.153594i	\(-0.0490848\pi\)
							−0.889697	+	0.456551i	\(0.849085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.p.a.146.6	yes	216
7.6	odd	2	inner	287.3.p.a.146.5	✓	216
41.25	even	10	inner	287.3.p.a.230.5	yes	216
287.230	odd	10	inner	287.3.p.a.230.6	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.p.a.146.5	✓	216	7.6	odd	2	inner
287.3.p.a.146.6	yes	216	1.1	even	1	trivial
287.3.p.a.230.5	yes	216	41.25	even	10	inner
287.3.p.a.230.6	yes	216	287.230	odd	10	inner